Finite Group DatabasebetaFinite Group DatabasebetaOrder 672.
188 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 188 conjugacy classes of subgroups — too many to draw. Only its 102 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C22×C3×Dic7 | C2 | semidirect |
| 3. C3×C22⋊Dic7 | C2 | semidirect |
| 4. C3×C4⋊Dic7 | C2 | semidirect |
| 5. C3×C4×Dic7 | C2 | semidirect |
| 6. D4×Dic7 | C3 | direct (N × H) |
| 8. C3×C7×D4 | C4 | semidirect |
| 15. C22×Dic7 | C2×C3 | semidirect |
| 16. C22⋊Dic7 | C2×C3 | semidirect |
| 17. C4⋊Dic7 | C2×C3 | semidirect |
| 18. C4×Dic7 | C2×C3 | semidirect |
| 22. C22×C3×C7 | C2×C4 | semidirect |
| 24. C3×C4×C7 | C2×C4 | semidirect |
| 27. C3×Dic7 | D4 | direct (N × H) |
| 29. C7×D4 | C3×C4 | semidirect |
| 47. C22×C7 | C2×C3×C4 | semidirect |
| 49. C4×C7 | C2×C3×C4 | semidirect |
| 52. Dic7 | C3×D4 | direct (N × H) |
| 54. C3×D4 | Dic7 | direct (N × H) |
| 60. C3×C7 | C4×D4 | semidirect |
| 72. C22×C3 | C2×Dic7 | semidirect |
| 77. C3×C4 | C2×Dic7 | semidirect |
| 79. D4 | C3×Dic7 | direct (N × H) |
| 85. C7 | C3×C4×D4 | semidirect |
| 92. C22 | C2×C3×Dic7 | semidirect |
| 97. C4 | C2×C3×Dic7 | semidirect |
| 98. C3 | D4×Dic7 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×C7×D4 | 2 |
| C22×C3×Dic7 | 2 |
| C3×C22⋊Dic7 | 2 |
| C3×C4⋊Dic7 | 2 |
| C3×C4×Dic7 | 2 |
| D4×Dic7 | 3 |
| C3×C4×D4 | 7 |